Econ 134A Midterm 1

1. Econ 134A Midterm 1 John Hartman Form A Summer 2019 Session B

2. Q1 • Aspen is about to receive $80,000 for a loan, to be completely paid back with 50 monthly payments (starting one month from today). The stated annual interest rate for his loan is 12%, compounded monthly. The amount of principal reduction for each month is constant. How much will the last payment be? • Principal reduction per month is 80,000/50 = 1600 • Monthly interest rate = 12%/12 = 1$ • Last payment is equal to the principal reduction of the last month plus the interest earned on that principal over the previous month, or 1% of 1600 ($16). • Final payment = $1600 + $16 = $1616

3. #2 • A stated annual interest rate of 10%, compounded five times per year, is equivalent to a stated annual interest rate of _____%, compounded continuously. • SAIR5times per year = 10% • EAIR = (1 + 0.1/5)^5 – 1 = 10.41% • SAIRcontinuous = ln(1.1041) = 9.90%

4. #3 • Maudry is borrowing $10,000 today to fund a new small business. She has an interest-only loan for the first 12 months, and her first payment will be in one month. (Note: With this interest-only loan, each payment exactly equals the amount of interest accrued in the previous month.) How much will each of the first 12 payments be if the effective annual interest rate is 10%? • First twelve payments only account for the interest accrued over the previous month, so each payment will be equal to $10,000 times the monthly interest rate • monthly rate = (1 + EAIR)^(1/12) – 1, EAIR = 10% • => monthly rate = 7.974% so the first twelve payments are $79.74

5. #4 • Warren is considering investing in a local business. He can invest $12,000 today and will receive annual payments of $1,000, forever, starting five years from today. What is the profitability index of this investment if his effective annual interest rate is 7%? • PV of benefits = (1000/0.07) x (1/1.07)^4 = 10,898.50 • general perpetuity formula, discounted by an extra four years • Profitability Index = 10,898.50/12,000 = 0.908

6. #5 • Gretta receives $500 every 6 months, starting 3 months from now. If her stated annual interest rate is 20%, compounded continuously, what is the present value of this perpetuity? • EAIR = e^(0.2) – 1 = 22.14% • six-month-rate = (1.2214)^(1/2) – 1 = 10.52% • three-month-rate = (1.2214)^(1/4) – 1 = 5.13% • PV = (500/0.1052) * (1.0513) = 4,997.92 • C/r would be the present value of the perpetuity starting in six months, but this perpetuity starts three months sooner than that so we need to multiply by 1 plus the three-month rate

7. #6 • Avery starts a new IRA account and deposits $1,000 each year starting today. The account pays 3% effective annual interest. How much will she have in this account immediately after the deposit is made 4 years from today? • The deposit made today will accrue interest at 3% for the next four years, the deposit made in one year will accrue interest for three years after that, etc. Thus the equation for this is the following: • V = 1000(1.03)^4 + 1000(1.03)^3 + 1000(1.03)^2 + 1000(1.03)^1 + 1000(1.03) = 5309.13 • Note: this may also be done with the annuity formula: • V = (1000/0.03) x (1 – 1/1.03^5) x (1.03)^5

8. #7 • If the real interest rate is 10% and the nominal rate of interest is 32%, the inflation rate is _____. • r = 0.1 • i = 0.32 • (1 + π) (1 + r) = 1 + i • π = (1 + i)/(1 + r) – 1 • = 1.32/1.1 – 1 • = 20%

9. #8 • Peador invests $10,000 today and receives a future value 20 years from now of $100,000. Interest is compounded monthly. The stated annual interest rate is _____, compounded monthly. • EAIR = (FV/PV)^(1/T) – 1 = (10)^(1/20) – 1 = 12.2% • SAIR = (1.122^(1/12) - 1) x 12 = 11.5%

10. #9 • Today is August 20, 2019. Assume that a coral tree can be purchased today for $100, and changes in the cost of a coral tree are the same as the rate of inflation. If annual inflation over the next two years is 10%, and 20% every year thereafter, what will be the cost of a coral tree on August 20, 2022? • Price today is $100. Over the next year, the price will increase by 10%. The price will then increase by 10% over the following year, and by 20% the year after that. • Thus the price in three years is equal to 100 x 1.1 x 1.1 x 1.2 = $145.2

11. #10 • What is the annuity factor of a 30-year annuity with an effective annual interest rate of 5%? Assume annual payments, with 30 payments starting one year from today. • Annuity factor = 1/r – (1/r)x(1/(1+r)^T) • In this case, the annuity factor is equal to • = 1/0.05 - (1/0.05)x(1/(1.05)^30) • = 15.37

12. #11 • An initial deposit of $1,000 turns into $1,590 ____ months from now if the stated annual interest rate is 6%, compounded monthly. • r = SAIR/12 = 0.5% • 1590 = 1000(1+r)^m where r is the monthly interest rate • 1590/1000 = 1.05^m • log(1590/1000) = m log(1.05) • 0.464 = m x 0.00499 • m = 92.98

13. #12 • A refrigerator requires a $700 purchase price today. It lasts for 7 years and has a single maintenance cost of $250 three years from today. What is the equivalent annual cost of the refrigerator if the effective annual interest rate is 2%? • Set the present value of the costs equal to the present value of an annuity with constant payment EAC • 700 + 250/(1.02)^3 = EAC/0.02 x (1 – (1/1.02)^7) • 935.58 = EAC x 6.472 • EAC = 144.56

14. #13 • Michelle will buy a $1,250,000 house today. She will make a 30% down payment, and borrow the remaining amount in a mortgage. She will repay the loan with 300 payments of $9,200, starting one month from today, plus a balloon payment in 310 months. How much will the balloon payment be in order to completely pay off the loan if the stated annual interest rate is 12%, compounded monthly? • Mortgage remaining = 1,250,000 x (1 – 0.30) = 875,000 • Monthly interest rate = 12%/12 = 1% or 0.01 • 875,000 = PV(annuity) + PV(balloon)

15. #13 cont. • 875,000 = PV(annuity) + PV(balloon) • PV(annuity) = 9200/0.01 x (1 – (1/1.01)^300) • PV(annuity) = 873,508.36 • PV(balloon) = 875,000 - 873,508.36 = 1,491.73 • PV(balloon) = X/(1.01)^310 • $X paid in 310 months • 1,491.73 x 1.01^310 = X • X = 1,491.73 x 21.86 = 32,607.36

16. #14 • The Constant Sorrow Tissue Company is considering opening a new plant in Hemet, CA. The cost of the new plant would have a present value of a $10 million cost if built. If the plant is built, it will generate net revenue of $1 million each of the next 50 years (starting in one year). Should the plant be built if the effective annual discount rate is 8%? Should the plant be built if the effective annual discount rate is 12%? Based on the answers to the two previous questions, is the internal rate of return for this new plant less than 8%, between 8-12%, or more than 12%? Justify your answer.

17. #14 cont. • PV(costs) = $10,000,000.00 in both cases • PV(benefits) = C/r x (1 – 1/(1+r)^50) • If r = 8% • PV(benefits) = 1/0.08 x (1 – 1/(1.08)^50) • PV(benefits) = 12,500,000 x 0.9787 • = 12,233,484.64 • NPV = PV(benefits) – PV(costs) = 12,233,484.64 - 10,000,000.00 • NPV = 2,233,484.64 • Thus, the plant should be constructed if r = 8%

18. #14 cont.(2) • PV(costs) = $10,000,000.00 in both cases • PV(benefits) = C/r x (1 – 1/(1+r)^50) • If r = 12% • PV(benefits) = 1/0.12 x (1 – 1/(1.12)^50) • PV(benefits) = 8,333,333.33 x 0.9965 • = 8,304,498.49 • NPV = PV(benefits) – PV(costs) = 8,304,498.49 - 10,000,000.00 • NPV = -1,695,501.51 • Thus, the plant should not be constructed if r = 12%

19. #14 cont.(3) • The internal rate of return is the discount rate ρ that sets the net present value of the investment equal to 0 • First note that the net present value is a strictly decreasing function of the discount rate (as r increases, NPV decreases because the all benefits and no costs are occur in the future) • The NPV when r = 0.08 is positive, so the IRR ρ must be greater than 8% • Similarly, the NPV when r = 0.12 is negative, so the IRR ρ must be less than 12% • Therefore, the internal rate of return is between 8% and 12%